A generalized framework for nodal first derivative summation-by-parts operators
A generalized framework for nodal first derivative summation-by-parts operators
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DOI：
10.1016/j.jcp.2014.01.038
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发表时间：
2014-06-01
影响因子：
4.1
通讯作者：
Zingg, David W.
中科院分区：
文献类型：
--
作者：
Fernandez, DavidC. Del Rey;Boom, Pieter D.;Zingg, David W.
提出了一个广义框架，扩展了经典理论的有限差分求和部分（SBP）运营商，包括广泛的运营商，其中主要的扩展是（i）非重复的内点运营商，（ii）非均匀节点分布在计算域，（iii）运营商，不包括一个或两个边界节点。证明了一阶导数具有SBP性质的节点逼近存在的充分必要条件。证明了每个SBP算子的正定范数矩阵必须与一个求积规则相关联，并且给定一个求积规则存在相应的SBP算子，其中对于对角范数SBP算子，求积规则的权必须为正.广义框架给出了一个简单的方法，提出了许多已知的近似SBP算子的一阶导数，几个调查，如不连续的Galerkin离散的基础上的Legendre-高斯积分点，并显示SBP算子。此外，新的框架提供了一种方法，用于构建SBP运营商从正交规则开始，这是通过构建新的SBP运营商从已知的正交规则。为了证明推广的效用，Legendre-Gauss和Legendre-Gauss-Radau求积点被用来构造不包括一个或两个边界节点的SBP算子。(C)2014爱思唯尔公司All rights reserved.
A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are (i) non-repeating interior point operators, (ii) nonuniform nodal distribution in the computational domain, (iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the SBP property. It is proven that the positive-definite norm matrix of each SBP operator must be associated with a quadrature rule; moreover, given a quadrature rule there exists a corresponding SBP operator, where for diagonal-norm SBP operators the weights of the quadrature rule must be positive. The generalized framework gives a straightforward means of posing many known approximations to the first derivative as SBP operators; several are surveyed, such as discontinuous Galerkin discretizations based on the Legendre-Gauss quadrature points, and shown to be SBP operators. Moreover, the new framework provides a method for constructing SBP operators by starting from quadrature rules; this is illustrated by constructing novel SBP operators from known quadrature rules. To demonstrate the utility of the generalization, the Legendre-Gauss and Legendre-Gauss-Radau quadrature points are used to construct SBP operators that do not include one or both boundary nodes. (C) 2014 Elsevier Inc. All rights reserved.